Optimization of offshore natural gas field development

Transcription

1 Optimization of offshore natural gas field development Gaute Rannem Johansen Industrial Economics and Technology Management Submission date: June 2011 Supervisor: Mikael Rönnqvist, IØT Co-supervisor: Vidar Gunnerud, ITK Norwegian University of Science and Technology Department of Industrial Economics and Technology Management

2

3

4

5 Preface This Master Thesis is written as the final work completing the study of Industrial Economics and Technology Management at the Norwegian University of Science and Technology (NTNU). It combines the fields of operations research and petroleum technology, allowing a large range of the knowledge acquired during the author s studies to be used. I would like to thank supervisor Mikael Rönnqvist and co-supervisor Vidar Gunnerud for valuable input and advice during the development of the thesis work. Further, I would like to thank my friends on the study of Industrial Economics and Technology Management. Without you the world would be a much less entertaining place. Trondheim, June 11th, Gaute Rannem Johansen i

6 Abstract In this thesis the target is to find the optimal development solution of an offshore natural gas field. Natural gas is increasing in importance as an energy source. Whilst most of the large oil fields have been developed, there are still several major natural gas deposits that may be developed. In addition, there are also smaller offshore natural gas fields that may be put into production. Finding an optimal development solution for these resources will increase the availability of natural gas. The objective of the mathematical model presented in this thesis is to maximize the total net present value of an offshore natural gas field development. The model does this by finding the optimal combination of investment decisions of the necessary natural gas field infrastructure. Infrastructure included in the model includes the number of wells to be drilled, flowlines, production facilities, energy infrastructure on site and transport infrastructure to the customers. The model also decides whether gas sales agreements should be made with selected customers and the natural gas production in all time periods. This offshore natural gas field development problem is formulated as a mixed integer linear programming problem. Piecewise linearization is used to increase the accuracy of the reservoir model and to find the energy demand for a given natural gas flow rate and pressure difference. The linearization makes the model easier to solve than if it was formulated as a non-linear program. Branch and bound was chosen as the solution method for the implementation of the model. The model has been implemented in the Mosel programming language, using Xpress-MP as the solver. Results from testing of the model on three different test cases indicate promising potential for the utilisation of the model. Optimal solutions were found in less than six minutes for all of the test cases, and close to optimal solutions were found quickly in the global search. ii

7 Contents 1 Introduction History of natural gas production History of natural gas production in Norway Natural gas fundamentals Forming of natural gas Petroleum exploration Natural gas classifications Natural gas specifications Natural gas field development Wells Subsea manifolds, templates and flowlines Production facilities Transport infrastructure Energy infrastructure Other infrastructure Natural gas markets and field development Fluid mechanics Pressure drop in pipelines Previous work Theory Mathematical programming models Linear programming Integer programming Non-linear programming Mixed integer non-linear programming Mixed integer linear programming Conversion of MINLP problems to MILP problems Special ordered sets Branch and bound Mathematical model Assumptions and simplifications Reservoir model Wells Flowlines Production infrastructure Transport infrastructure Energy infrastructure iii

8 3.8 Customers Objective function Implementation Software Solution method Model structure Test cases Test case Test case Test case Results Hardware and software specifications Test Case Test Case Test Case Discussion 69 8 Conclusion 74 9 Further work 75 A Mathematical model 79 A.1 Sets and indices A.2 Data A.3 Variables A.4 Objective function A.5 Constraints B Mosel code 89 iv

9 List of Figures 1.1 Historical world natural gas production Oil and natural gas production in Norway Natural gas production in Norway Recipients of Norwegian gas exports by pipeline Recipients of Norwegian exports of LNG Huldra. A fixed platform Troll A platform Production profile for the Frigg natural gas field Production profile for the Huldra natural gas field Natural gas, NGL and condensate production profile for the Troll field p/z-plot for the Huldra natural gas field Mathematical model structure Natural gas production in Test Case Natural gas production in Test Case Natural gas production in Test Case Initial optimality gap of integer solutions of Test Case Initial optimality gap of integer solutions of Test Case Initial optimality gap of integer solutions of Test Case Objective function values of integer solutions of Test Case Objective function values of integer solutions of Test Case Objective function values of integer solutions of Test Case v

10 List of Tables 1.1 Natural gas composition of some natural gas fields Main properties of Test Case Infrastructure alternatives in Test Case Main properties of Test Case Infrastructure alternatives in Test Case Infrastructure alternatives in Test Case Main properties of Test Case Infrastructure alternatives in Test Case Infrastructure alternatives in Test Case Hardware and software specifications Results for Test Case Results for Test Case Results for Test Case Comparison of test cases vi

11 Nomenclature DCQ - Daily Contract Quantity FLNG - Floating LNG FPSO - Floating Production, Storage and Offloading IP - Integer Programming LNG - Liquefied Natural Gas LP - Linear Programming MEG - Monoethyleneglycol MILP - Mixed Integer Linear Programming MINLP - Mixed Integer Nonlinear Programming MIP - Mixed Integer Programming NCS - Norwegian Continental Shelf NGL - Natural Gas Liquids NPD - Norwegian Petroleum Directorate NPV - Net Present Value NLP - Non-Linear Programming PI - Productivity Index PIP - Pure Integer Programming SOS1 - Special Ordered Set of type 1 SOS2 - Special Ordered Set of type 2 TLP - Tension Leg Platform WI - Wobbe Index vii

12 1 Introduction 1.1 History of natural gas production Mankind has known natural gas for thousands of years. The temple in Greece where the famous Oracle of Delphi lived was built where seepage of natural gas from underground created a burning flame. This occurred approximately 1000 B.C. [NaturalGas, 2011a]. Around 500 B.C the Chinese built pipelines out of bamboo shoots in order to utilise the natural gas. The natural gas was used to boil saltwater, making it drinkable. In the late 18th and early 19th century gas manufactured from coal, so-called coal gas, was being used for street lighting in Britain. This is one of the first major utilisations of natural gas. In 1821 the first intentional natural gas well in the United States was dug in Fredonia, New York, when a man named William Hart dug a 27-foot well where natural gas was seeping to the surface to obtain a higher flow rate. Already in 1869 the first offshore drilling rig patent was given. Despite this, onshore fields would remain the main natural gas fields for the next century. In this early period of petroleum production natural gas was not considered particularly useful. Although some pipelines were built, they were not very efficient. Due to this lack of transport alternatives, natural gas was usually vented to the atmosphere, burnt on site or left in the ground when found alone before World War II. In the 1920s an effort was put into building a pipeline infrastructure in the United States. After World War II advances in pipeline technology allowed an extensive pipeline network to be built in the United States. This opened new possibilities for utilisation of natural gas in the United States. Figure 1.1 shows the total natural gas production in the world from 1970 to 2009, using the data from the BP Statistical Review of World Energy June 2010 [BP, 2010]. The natural gas production has been steadily increasing, until In 2009 the global natural gas production declined for the first time on record. This was largely due to declining consumption because of the financial crisis. 1.2 History of natural gas production in Norway In 1959 the giant Groningen gas field was discovered in the Netherlands [Whaley, 2011]. This made oil companies interested in exploration activities in the North Sea. The Norwegian government were surprised when they were approached by 1

13 3,500 World natural gas production Billion cubic meters 3,000 2,500 2,000 1,500 1, Year Figure 1.1: Historical world natural gas production [BP, 2010] Phillips Petroleum Company, asking for the rights of exploration activities on the Norwegian Continental Shelf (NCS). In 1958 Norges geologiske undersøkelse had announced in a letter to the Ministry of Foreign Affairs that it was unlikely that oil could be found by the Norwegian coast [NRK, 2011]. Oil and gas exploration activities in Norway commenced in the 1960s. The first oil field was discovered in 1967, when the Balder field was found. However, this field was not considered profitable back then and it was not developed until 30 years later. In 1969 the first giant field, Ekofisk, was discovered by Phillips Petroleum Company. Petroleum production on the NCS started in 1971 from this field. More giant discoveries followed in the 1970s, such as Statfjord, Gullfaks, Oseberg and Troll. The first major natural gas find was the Frigg field, which was found in 1971 [Tønnesen, 2011]. It was the world s largest offshore gas field when it was discovered [Frigg, 2011]. As these fields were brought on stream throughout the 80s and 90s, Norway rapidly became a major oil and natural gas producer, as can be seen in Figure 1.2 which shows the historical petroleum production on the Norwegian Continental Shelf. These fields found in the North Sea in the 70s have been the backbone of the Norwegian petroleum production. Exploration in the Norwegian Sea and Barents Sea followed in the 1980s, with several major discoveries. An extensive network of oil and gas pipelines have been built to exploit these resources, as well as the 2

14 Net - condensate [mill Sm3] Net - NGL [mill Sm3] Net - gas [bill Sm3] Net - oil [mill Sm3] Figure 1.2: Oil and natural gas production in Norway [NPD, 2011] northernmost liquefied natural gas (LNG) plant in the world at Melkøya outside Hammerfest. Natural gas exports from Norway started in 1977 when the Norpipe pipeline from the Ekofisk field to Germany was opened. The second field that started exporting natural gas from Norway was the Frigg field. Production started from the British part of this field in 1977, and from the Norwegian part in The gas was transported by the Vesterled pipeline to St. Fergus in Scotland. Following the development of these two pipelines, the Norwegian natural gas production remained relatively stable at a production level of around 25 billion standard cubic meters per year until the mid 1990s. The historical Norwegian natural gas production is illustrated in Figure 1.3. The giant Troll field was discovered in 1979 by Norske Shell. This field contains more than 40 per cent of the total gas reserves on the NCS [Statoil, 2011]. The development of this and other natural gas fields in the late 1990s led to a significant increase in the total natural gas production in Norway, as seen in Figure 1.3. The natural gas production in Norway is mainly exported to the United Kingdom and continental Europe by pipelines. Receiving terminals have been built in the United Kingdom, France, Belgium and Germany. From these terminals the natural gas may end up in other countries after being transported through the existing distribution network. The distribution of where Norwegian gas exported by pipeline is sold is illustrated with Figure

15 Net - gas [bill Sm3] Figure 1.3: Natural gas production in Norway [NPD, 2011] 0% 2% 8% 25% 6% 1% 7% 31% 3% 17% Austria Belgium Czech Republic France Germany Italy Netherlands Spain Switzerland Figure 1.4: Recipients of Norwegian gas exports by pipeline [BP, 2010] 4

16 The construction of the LNG plant at Melkøya outside Hammerfest has allowed natural gas from the Snøhvit field to be produced. This natural gas is mainly exported to Spain and the United States, although other countries also buy some of this natural gas supply. Figure 1.5 illustrates where LNG exports from Norway are sold. 8% 27% US Mexico Belgium 43% 5% 3% France Spain 14% United Kingdom Figure 1.5: Recipients of Norwegian exports of LNG [BP, 2010] In recent years no giant fields have been found, with Ormen Lange in 1997 being the last giant discovery. However, exploration activity is still high, with a record high number of 65 exploration wells being drilled in In 2009 oil and gas companies invested about 136 billion NOK in field developments, exploration, transport infrastructure and onshore process facilities. Today 22 per cent of Norway s gross domestical product comes from the oil and gas industry. Oil and natural gas is responsible for approximately half of the total value of exports from Norway. In 2008 Norway was the world s second largest exporter of natural gas and the sixth largest exporter of oil. In 2010 the natural gas production surpassed the oil production for the first time in Norway. It is expected that Norwegian oil production will continue to decline, but the total petroleum production will be relatively stable as natural gas production is still increasing. Concerns have however been raised about Norway s ability to sustain natural gas production after 2020 [Lindeberg, 2011]. More natural gas discoveries are needed to maintain a high level of natural gas exports to Europe. Today many fields are developed as tie-backs to existing infrastructure. 5 Some

17 examples of this is the Hyme development that will connect to the Njord platform, and Morvin that is tied-back to Åsgard. This is in contrast to the large concrete platforms that were used to develop the old, giant fields such as Frigg and Troll. Cooperation between different fields is important to develop marginal resources. Utilising existing infrastructure also contributes to increased petroleum recovery from old fields, as the petroleum from new fields makes it profitable to continue production from the old fields. Finding an optimal development plan is increasingly important to make these marginal projects profitable. 1.3 Natural gas fundamentals Natural gas is an odourless mixture of light hydrocarbons, water, nitrogen, hydrogen sulphide, carbon dioxide and other trace components. The main component in natural gas is methane (CH 4 ). Natural gas is the cleanest fossil fuel when comparing CO 2, NO x, SO 2 and particulate emissions on a per energy unit basis [NaturalGas, 2011b]. The composition of natural gas varies from field to field, and can show large differences. Table 1.1 shows examples of the gas composition of natural gas from four different natural gas fields. Table 1.1: Natural gas composition of some natural gas fields [Gudmundsson, 2010] Molecule Troll, Sleipner, Draugen, Groningen, fraction Norway Norway Norway Netherlands Methane Ethane Propane Iso-Butane N-Butane C Nitrogen Carbon dioxide The behaviour of natural gas as the pressure and temperature changes is often visualised with a so-called phase envelope. The phase envelope describes which phase a natural gas mixture is in at a given pressure and temperature. This concept will not be discussed further in this thesis, but two properties related to the phase envelope need to be mentioned. These are the cricondenbar pressure and the cricondentherm temperature. The cricondenbar is the highest pressure on the phase envelope curve. Above this 6

18 pressure the natural gas will be in a single dense phase at all temperatures. If the temperature is lowered the gas mixture will become more dense and liquid-like, but no phase transition will happen and no condensation of liquid will occur. Similarly, the cricondentherm is the highest temperature on the phase envelope. As long as the temperature of the natural gas mixture is above the cricondentherm, it will never be in liquid phase. These two properties are important related to natural gas specifications, which are described in Chapter Forming of natural gas Natural gas is formed during a long process, lasting millions of years. Several conditions need to be met if natural gas is to be formed. Exploration for natural gas involves looking for three types of rock: source rock, reservoir rock and cap rock. These rocks are necessary for the formation of a natural gas reservoir. The following explanation of how natural gas is formed is mainly based on Morgan Downey s book on oil [Downey, 2009]. Source rock is rock laden with kerogen, which is a solid dark waxy rock. Kerogen is the portion of naturally occurring organic matter that is nonextractable using organic solvents [Schlumberger, 2011]. Kerogen typically constitutes of algae and plant material. It is mostly formed by sedimentation of marine material. An example of a source rock is the Kimmeridge Clay (Draupneskifer in Norwegian). Kimmeridge Clay is the main source rock of the oil deposits in the North Sea [SNL, 2011]. If this source rock is buried deep enough, the heavy hydrocarbon molecules in the kerogen will, over time, be cracked into lighter hydrocarbons. These lighter hydrocarbons are oil and natural gas. This process where crude oil and natural gas are formed from the source rock is called maturation. The amount of heat that is necessary to form oil and natural gas from kerogen is reduced when the pressure increases. Both the temperature and pressure increases with depth underground. Thus, the deeper the kerogen is buried, the higher the temperature and pressure, and the lighter the oil that is formed will be. If the kerogen is buried even deeper, all the oil will be cracked into methane. The deeper level where all crude oil formed at shallower depths is cracked into methane is known as the gas window. How deep this level is varies around the world. When the less dense crude oil and natural gas are formed, it will migrate upwards towards the surface. If no suitable reservoir rock and cap rock are in place, the oil and gas will migrate all the way to the surface. Here it will evaporate, be eaten 7

19 by bacteria or oxidize. Most of the oil that has been formed throughout history has escaped in this manner. The reservoir rock needs to be in contact with the source rock and be porous enough to contain the natural gas. Contrary to common perception, oil and natural gas are not found in pool-like domes, but occupy the pores of sedimentary rocks. The two main types of reservoir rocks are sandstone and carbonates. Both of these are in place on the NCS. For example, Ekofisk is a carbonate reservoir, while Statfjord is a sandstone reservoir. If petroleum production shall be feasible, the reservoir rock also needs to be permeable enough to let the oil and gas pass through to a production well. Finally, a cap rock is needed to keep the oil and gas in place. This cap rock may for example be shale, salt or anhydrite. The cap rock also needs to have a shape that can prevent the oil and gas from moving upwards. Some examples of this are anticline and fold traps, although other possibilities exist. 1.5 Petroleum exploration Previously, the fundamentals about how natural gas is formed and the necessary conditions for a natural gas reservoir to exist was outlined. Petroleum exploration is the activity where one tries to identify and find out where these reservoirs are, and how much petroleum a reservoir contains. Several geophysical methods might be used to try to get know ledge of the conditions underground. These include gravity surveys, magnetic surveys, electro-magnetic surveys and seismic surveys, of which seismic imaging is the most well-known. By analyzing data from these surveys oil companies try to identify possible reservoirs. Confirming that a prospect from these analyses contains oil and/or natural requires a well to be drilled. If petroleum is found, additional wells might be drilled to get more information about the size of the reservoir. Exploration activities are expensive, in particular exploration drilling. 1.6 Natural gas classifications Natural gas may occur as associated gas or non-associated gas. Associated gas is found together with oil, either as a gas cap or dissolved in the oil. Non-associated gas is natural gas with no underlying oil column. All the oil that has been formed has been cracked into gas for these kinds of reservoirs. This means that nonassociated gas usually has a higher content of methane and lower fraction of heavier hydrocarbons than associated gas. 8

20 Natural gas can also be divided into sweet gas and sour gas. These terms are related to the content of hydrogen sulphide and carbon dioxide in the gas. Sour gas has a high content of at least one of these components. Typical limits are 2 volume percent for carbon dioxide and 1 volume percent for hydrogen sulphide for the gas to be deemed sour gas. Sour gas needs extra treatment to remove these components as they are highly corrosive. The natural gas that has been found on the NCS is mostly sweet gas. Some sour gas fields exist on the NCS, an example is Sleipner from Table 1.1 and Snøhvit, which both have a high carbon dioxide content by Norwegian standards. Examples exist of high hydrogen sulphide or carbon dioxide content in natural gas. In Canada some wells produce natural gas with more than 30 percent H 2 S content [Jahn et al., 2008], and the Natuna gas field in Indonesia contains as much as 70 percent CO 2 [Technology, 2011]. In addition to being divided into associated or non-associated gas and sweet or sour gas, natural gas is also categorised as being dry or wet gas. Dry gas is gas with a very high content of methane, which will never condense into liquid at normal ambient conditions. Sales gas is dry gas, as the possibility of liquid formation in the end-customers burners can be dangerous. Wet gas, sometimes used interchangeably with rich gas, contains a larger amount of heavier hydrocarbons. These may condense during pipeline transport, processing or at the end user. Thus it needs further processing before being sold. 1.7 Natural gas specifications Processing of natural gas is done to meet some specification, which may vary depending on in which part of the value chain the processing is done. Transport specifications are necessary to ensure safe and reliable transport of natural gas from an offshore production facility to a processing plant for further processing. These are typically related to water content, outlet pressure at the platform and sour gas content. The water content must be low enough to make sure that no liquid water will drop out during worst case conditions. This is at high pressure and low temperature, which can happen if the export pipeline is blown down. The water content specification is given as a water dew point in terms of pressure and temperature. Necessary outlet pressure at the production facility is related to two factors. The outlet pressure must make sure that the natural gas is at the correct inlet pressure of the processing plant. Further, this must also make sure that the entire transport 9

21 happens above the cricondenbar of the natural gas. This is assured by previous processing at the production facility. It is necessary to transport the gas above the cricondenbar pressure to avoid possible liquid hydrocarbon dropout in the pipeline It is important to avoid liquid dropout in large parts of the value chain for several reasons. Hydrates may be formed, corrosion problems may occur and operational difficulties leading to safety risk may be experienced if liquids do form. The sour gas transportation specification is mainly dependent on two factors. First of all it is important for corrosion control of the pipeline. Elevated levels of carbon dioxide or hydrogen sulphide may easily corrode the pipeline, especially if free water is present. Secondly, it is related to sales gas specifications. The amount of sour gas removal that is required due to this is also dependent on two external factors. An amount of sour natural gas that would otherwise need extensive sour gas removal may avoid this if mixing it with sweet natural gas from another field results in natural gas that still meets sales specifications. Some processing plants also contain a sour gas removal unit, lowering the need of sour gas removal offshore. Sales specifications are important to ensure safe use of the natural gas for the end customer. They can be broadly divided into four areas. The energy content of the natural gas is an important measure for billing purposes. The customer is interested in getting the correct energy amount, and the gas seller wants to get paid for supplying it. Gross calorific value is used as a measurement of the energy content in the natural gas. Wobbe Index (WI) is used to measure the energy load of the natural gas. It is the main indicator of the interchange ability of different natural gas mixtures. Different countries often have different WI requirements. Combustion characteristics describe the combustion kinetics during combustion of the natural gas. Two factors describe this. The soot index gives an indication of the risk of soot formation during combustion. The Incomplete combustion factor indicates the risk of carbon monoxide formation during the combustion process. Finally there are specifications regarding impurities in the natural gas. These include for example mercury, water, nitrogen, hydrogen sulphide and carbon dioxide. 10

22 1.8 Natural gas field development After an oil company has found a natural gas field the next task is to find the best way to develop that natural gas field. Many decisions have to be made, and one decision is likely to affect the outcome of other decisions. It is important to try to look upon the field development as a whole, rather than choosing the optimal solution for each problem separately. Otherwise it is likely that the total solution will be suboptimal. This in turn makes optimization and operations research an interesting tool to find a good solution for the whole field development problem. The following subsections will outline the different parts of a natural gas field development Wells Wells need to be drilled if one wishes to produce the natural gas reserves in a given reservoir. Some production facilities have an integrated drilling facility to allow relatively easy drilling at all times. Other production platforms are dependent on moveable drilling rigs for the drilling of wells. The latter is often more expensive than the former. There are also different types of wells. The simplest type of well is the vertical well. These are drilled vertically down, and have a relatively short section where oil and gas is able to flow into the well. One way to reach more oil or gas is to drill horizontal wells. Technology advancements led to these becoming successful; as well trajectories could be accurately planned and followed. An even more advanced type of well is the multilateral well. This kind of well involves drilling several holes branching from a central borehole. These can be particularly attractive to drain remaining pockets of hydrocarbons in mature fields, and in subsea developments [Jahn et al., 2008]. Well costs can be a significant part of the investment costs of a natural gas field development. The day rate of a drilling rig may be around 2 million NOK/day, and even more for an advanced deep water drilling rig. Drilling a well may also take a long time, some exploration wells as long as three months [Salthe, 2011] Subsea manifolds, templates and flowlines Today, many fields on the NCS are developed as subsea field developments. Here, the equipment which is necessary if such a solution is chosen will be discussed. If the field is developed solely with wells drilled from a fixed platform, the wells can be connected to the production facility without the need for the following equipment. 11

23 The simplest form of subsea developments is with a single satellite well, connected to a production facility through a series of pipelines and umbilicals. This solution is typically chosen if the field to be developed is small and near a large field. If the field is large enough to justify the drilling of several wells, it will soon become uneconomical to tie each well back to the production facility with separate flowlines. Subsea templates and manifolds are used in this case. The subsea production template is generally recommended for use with six or more wells. It is commonly used when an operator has a firm idea of the number of wells that will be drilled in a certain location. All subsea facilities are contained within one protective structure [Jahn et al., 2008]. The subsea template is connected to the production facility with flowlines and umbilicals. If several single wells or templates need to be connected to the production facility, it might be economical to use an underwater manifold system. Each well or template are connected to this manifold, which is then tied back to the production facility with a single set of pipelines and umbilicals. This saves expenses on flowlines and umbilicals that would otherwise be required. Large field developments may involve a large network of umbilicals and flowlines, in order to get the best solution. In addition to flowlines bringing the natural gas towards the production facility, flowlines are also necessary to bring various chemicals that need to be injected into the natural gas flow. An example of this is the need to inject monoethyleneglycol (MEG) to avoid the formation of hydrates that would otherwise block the natural gas flow in the flowline. Other chemicals are also used, to avoid scaling and other issues Production facilities Natural gas needs to undergo several treatments in order to meet a given transport or sales specification. At least parts of this happens at a production facility. The following description of different production facilities is mainly based on Jahn et al. [2008]s book on hydrocarbon exploration and production. Production facilities come in a wide range of variations. Depending on local conditions such as water depth, distance to shore and already existing nearby production infrastructure, several different types of platforms might be considered. Production platforms can be divided into two main types: fixed and floating platforms. Fixed platforms can be used in relatively shallow water depths. The most common fixed platforms are steel jacket platforms and gravity-based platforms. Steel jacket platforms have been successfully used in both shallow and calm water, as well as 12

24 in demanding North Sea conditions. Many fields on the NCS have platforms of this type, such as the Ekofisk and Valhall fields. Steel jacket platforms are used in water depths of up to 150 m [Jahn et al., 2008]. Figure 1.6 shows an example of a fixed steel jacket platform. Figure 1.6: Huldra. A fixed platform. Photo: Kjetil Alsvik/Statoil The concrete gravity-based platform of the Condeep type is possibly the bestknown platform of the Norwegian public. Many of the largest fields on the NCS have been developed with one or more platforms of this type. Troll, Statfjord and Gullfaks are some of the fields that have at least one Condeep platform. Concrete platforms can be used in similar water depths as steel jacket platforms, and slightly deeper. The water depth around the Troll A platform, shown in Figure 1.7, is over 300 metres. A field does not need to be developed with only one sort of platform. Indeed, both steel jacket and gravity-based have several times been used together to develop a field. For example, a gravity-based platform may hold heavy processing facilities whilst a steel jacket platform may be used as a wellhead platform. As the water depth increases fixed platforms become infeasible. Floating platforms are able to operate in much greater water depths than fixed platforms. Many variations of the floating platforms exist, of which some will be described here. 13

25 Figure 1.7: Troll A platform. Photo: Øyvind Hagen/Statoil Semi-submersible platforms have been used in water depths of over 1800 m. Pontoons at the base of the platform construction are filled with water, partially submerging the platform and thus giving it its name. The platform is moored to the bottom with anchor chains. Semi-submersible platforms have been widely used on the NCS. Snorre, Kristin and Gjøa are some of the fields that have been developed with semi-submersible platforms. Floating storage, production and offloading vessels (FPSOs) are another possible development solution. They have been used in water depths of up to 2600 m. The biggest FPSO on the NCS to this date is expected to start producing in 2011, when the Skarv FPSO comes on stream. Skarv is a natural gas and oil field, with a majority of natural gas. Several other fields on the NCS have been developed with FPSOs, for example Norne and Åsgard. SPAR platforms resemble large cylinders supporting a platform. These can be used in very deep water. They are tethered to the bottom with cables and lines. SPARs have been very popular in the Gulf of Mexico, but to the author s knowledge no SPAR platform is installed in Norway. The current plan for the development of the Luva natural gas field does however involve building the largest SPAR in the world. 14

26 Tension leg platforms (TLPs) fall somewhat in between fixed and floating platforms. They consist of a floating platform that is attached to the sea floor by long tension legs, allowing some movement from side to side, but keeping the platform relatively stable vertically. The Heidrun field is developed with a TLP. Subsea production systems have been discussed earlier. Advances in multiphase technology have made it possible to develop a field with only subsea structures offshore in some cases. These are connected to an onshore processing plant by pipelines. All the processing happens at this plant. Ormen Lange is an impressive example of a subsea development. Located 120 km offshore at a water depth of 850 to 1100 m, the gas is processed at the Nyhamna plant, before being exported to the United Kingdom. The Snøhvit field is also a subsea development. Finally, plans for floating LNG (FLNG) plants are currently being made by several oil companies. Recently, Shell announced that they have made the final investment decision on the Prelude FLNG project, making it the world s first FLNG facility [Shell, 2011]. FLNG plants resemble FPSOs, by having all gas processing and gas liquefaction at one vessel offshore. FLNG will make it possible to develop gas resources that were previously considered to be too far away from land for development Transport infrastructure After some processing at the production facility the natural gas will be transported either to an onshore processing plant for further processing or directly to the customer. This depends on several factors such as gas composition and distance to the customer. There are two main ways of transporting the gas: through a gas pipeline or as LNG. The choice of how to transport the gas is mainly a matter of distance to the market, even if LNG offers some additional flexibility regarding where to transport the gas. The transportation cost for LNG is characterised by that the transportation cost for short distances is very high compared to pipeline transport, but that the costs increase much slower than for pipeline transport as the transport distance increases. Over a certain distance LNG is the cheaper option. During the past 40 years Norway has developed the world s largest system of underwater pipelines. This means that it will often be profitable to use this pipeline system, rather than building a new export pipeline to transport the natural gas to the market. The cost for transportation of natural gas is then reduced to an investment cost and operational cost for a new, smaller pipeline connecting to the existing pipeline and a tariff for using the existing pipeline. 15

27 If the natural gas field that is to be developed is large enough, or the capacity of the existing pipeline system is too small for a long enough time, it will anyway be necessary to build a new pipeline. Specialised pipeline laying vessels are used to do this Energy infrastructure A production platform contains a lot of power consuming equipment. Pumps and compressors used for natural gas processing and other purposes can have very high power demand. This power needs to be acquired either by generating it on the platform or by importing it from another source. Most natural gas field developments until now have generated the necessary power internally on the platform by the use of gas turbines. The necessary amount of fuel gas is taken from the produced natural gas after some processing, usually before the last step of compression for exports. There are however cases where electricity is imported to the production facility. Several reasons might make this an attractive solution. In Norway there has been a political desire from the government that electricity should be supplied from the onshore grid, in an effort to reduce carbon dioxide emissions. It has often been required to investigate the possibility of importing electricity during the planning process of a field development. Weight limitations on the production platform may also make import of electricity interesting. If there is a tax on carbon dioxide emissions, low electricity prices and high natural gas prices, importing electricity may become a cheaper solution than generating power by gas turbines. Subsea developments also need to import electricity, as one obviously cannot use gas turbines underwater. Several field developments import electricity to cover some or all of the power demand. This includes amongst other the Troll and Gjøa natural gas field developments. The Goliat field in the Barents Sea will import about half of the power demand, and the Valhall field redevelopment includes electrification of the platforms. Limited availability of power may restrict the field development. The maximum size of a gas turbine might for example be limited by weight or available space. It is also possible that nearby platforms or the onshore grid has a limited amount of available power. 16

28 1.8.6 Other infrastructure There are many other decisions that have to be taken in addition to those regarding equipment that has been described previously. An important issue is how to avoid the formation of hydrates. It is possible to avoid this in several ways. Perhaps the most common hydrate prevention system in use on the NCS is the injection of MEG into the well stream. Other systems include methanol injection and electrical heating systems. Usually some oil or condensate is produced together with the natural gas. These liquid products need to be stored, transported to the market and sold. The condensate may be transported with ships to the market, or by a dedicated condensate pipeline. The processing system on the production facility contains several different pieces of equipment. Decisions need to be made for example regarding the use of turbines or Joule-Thomson valves to reduce pressure, what kind of separators that are to be used and how many compression steps that are needed. 1.9 Natural gas markets and field development Natural gas field developments have usually required a long plateau production period compared to oil field developments. A gas field typically has a plateau production period of around 10 years, producing around two thirds of the reserves on plateau production. This can be compared to typical oil field plateau production period of 2-5 years. Three examples of natural gas production profiles are shown in Figure 1.8, 1.9 and In these examples, Frigg has been closed down whilst Huldra and Troll are still producing. Whereas a spot market has always existed for oil, gas sales traditionally require a contract to be agreed between the producer and a customer. This forms an important part of gas field development planning, since the price agreed between producer and customer will vary, and will depend on the quantity supplied, the plateau length and the flexibility of supply. Whereas oil price is approximately the same across the globe, gas prices can vary very significantly from region to region. To attain a good sales price for the gas the customer usually requires a reliable supply of gas at an agreed rate over many years. Long term contracts are often linked to the oil price. An example of a long term contract is the Troll gas sales agreement, in which gas sales for a period of 30 years were agreed. 17

29 Net - condensate [mill Sm3] Net - gas [bill Sm3] Figure 1.8: Production profile for the Frigg natural gas field [NPD, 2011] Net - NGL [mill Sm3] Net - oil [mill Sm3] Net - gas [bill Sm3] Figure 1.9: Production profile for the Huldra natural gas field [NPD, 2011] 18

30 Net - condensate [mill Sm3] Net - NGL [mill Sm3] Net - gas [bill Sm3] Figure 1.10: Natural gas, NGL and condensate production profile for the Troll field [NPD, 2011] These contracts do not mean that production is completely stable over the whole contract period. Instead, a set of parameters are agreed upon such as: Daily contract quantity - The daily production which will be supplied, (DCQ) usually averaged over a period such as a quarter year Swing factor - The amount by which the supply must exceed the DCQ if the customer so requests Take or pay agreement - If the buyer chooses not to accept a specified quantity, he will pay the supplier anyway Penalty clause - The penalty which the supplier will pay if he fails to deliver the quantity specified within the DCQ and swing factor agreements Although the natural gas spot market is increasing in importance in Europe, and more fields are developed without long term contracts, most of the gas on the NCS is still sold on long term agreements. Statoil sell almost 90 per cent of their natural gas on long term contracts linked to the oil price [Steensen, 2011]. The natural gas market in Europe has been increasingly deregulated, both at the distribution side in continental Europe and at the transport side. 19

31 1.10 Fluid mechanics In order to determine the natural gas flow rates, one has to perform analyses with tools from the field of fluid mechanics. This section will present some of the most important relations in fluid mechanics. The flow rate of any fluid is defined by equation (1.1): Q = A V nda (1.1) Where Q is the volumetric flow rate, V is the velocity, n is the normal vector of the surface, da is the surface differential and A is the area over of which the flow rate is calculated. The real gas law is especially important to describe the behaviour of natural gas when the pressure changes. pv = zrt (1.2) Where p is the pressure, v is the specific volume of the gas, z is the compressibility factor, R is the universal gas constant and T is temperature. As the gas volume is dependent on temperature and pressure, the volume is often given at standard conditions. These standard conditions may actually vary, so it is essential to check what the standard conditions are for a given volume. An example of a standard condition can be 15 degrees C and 1 atm pressure. The relationship between volume at standard conditions and volume at some other condition is given as: V s.c. = V ( p p s.c. ) ( Ts.c. T ) ( ) 1 z (1.3) The gas flow equation for low pressure steady state flow can be shown to be: q s.c. = πkht s.c(p 2 R p2 w) p s.c. T µ z[ln re r w 1 2 ] (1.4) Where: 20

32 q s.c. - flowrate at standard conditions, m 3 /s k - permeability of the reservoir rock, m 2 h - height of producing formation, m T s.c. - temperature at standard conditions, K p R - reservoir pressure, P a p w - well (flowing) pressure, P a p s.c. - pressure at standard conditions, P a T - reservoir temperature, K µ - fluid viscosity at average pressure, P as z - fluid compressibility at average pressure, 1 r e - distance to drainage boundary, m r w - wellbore radius, m A productivity index, PI, can be defined by rephrasing this equation. As long as the viscosity and temperature remains constant, this productivity index will also remain constant. P I = πkht s.c p s.c. T µ z[ln re r w 1] (1.5) 2 By using this formulation, a simpler form of the gas flow equation can be given. q s.c. = P I(p 2 R p 2 w) (1.6) An even simpler gas flow equation is appropriate at high pressure conditions. At high pressure, the gas will behave in a similar way to liquids such as oil. The gas flow rate is then dependent on the pressure difference p R p w, which is linear. q s.c. = P I(p R p w ) (1.7) Producing natural gas will lead to a pressure reduction in the reservoir. This will again limit the production potential of the reservoir. The pressure drop in the reservoir is linked to the production in an almost linear manner. The non-linearity occurs due to the z-factor of the gas changing because of the reduction in pressure. Equation (1.8) shows the relationship between the pressure p, compressibility z, initial gas resource G i, initial pressure p i, initial gas compressibility z i and cumulative natural gas production, G. ( p ) = z ( pi z i ) (1 GGi ) (1.8) 21

33 This is often illustrated with a p/z-plot, which can have remarkably good fit to actual data as can be seen in Figure Figure 1.11: p/z-plot for the Huldra natural gas field [Saksvik, 2004] Pressure drop in pipelines Natural gas field developments usually contain several pipelines, in the form of flowlines and export pipelines. The pressure drop in these pipelines will affect the outcome of a project. Compressing natural gas is expensive energy-wise, so having low pressure loss in the export pipelines is important. Also, one wants to be able to keep the wellhead pressure as low as possible to achieve as high a production rate as possible. This means that the pressure loss in the flowlines should be low. The total pressure loss in a pipeline is due to three effects: hydrostatic difference, acceleration and friction. As the hydrostatic pressure loss is given of external conditions and acceleration can be avoided in steady state conditions, the pressure loss because of friction is the most important of these three effects regarding optimization. p = p g + p a + p f (1.9) p g = ρg sin α L (1.10) p a = ρu u (1.11) p f = f L 1 d 2 ρu2 (1.12) 22

34 The equation for pressure drop due to friction, p f is known as the Darcy-Weissbach equation. This means that it will usually be favourable to have a large diameter pipeline. It is necessary to weigh this up against increased investment cost and, in the case of gas-condensate being transported, the liquid volume build-up in the pipeline. Another way to reduce the pressure loss in the pipeline is by reducing the friction factor f. Assuming that the Reynolds number of the gas is higher than 10 5, this needs to be done by using a smoother pipe. This will be more expensive, but may be profitable due to the reduced pressure drop in the pipeline Previous work Optimizing offshore oil and gas field developments have had the attention of oil companies for a long time. Different authors have approached the problem in various ways. Frair and Devine [1975] formulate a deterministic MIP model where the number of platforms, platform capacity and location, assignment of wells to platforms, platform placing and well drilling schedule and production rate in each time period are determined. The number of platforms is fixed in the model, and then the model is solved for several numbers of platforms to determine the optimal number. Maximizing the after-tax profit is the objective in this model. In order to solve the model it is partitioned into one platform location subproblem and a well drilling, platform placement and reservoir production scheduling problem. These subproblems are then solved sequentially. McFarland et al. [1984] formulate a deterministic non-linear programming model where the decision variables are the number of wells to be drilled, production rates, abandonment time and platform size. The reservoir is modelled as a zerodimensional tank. A gas reservoir with water drive and a three phase oil reservoir are modelled. Reduced gradient methods are used to solve this non-linear model. The flow rates are modelled as being pressure dependent. Haugland et al. [1988] present a MILP model that can propose platform capacity, where and when to drill wells and production from the wells in each time period. These decisions are taken simultaneously, as opposed to Frair and Devine s sequential approach. A two-dimensional reservoir model is used, and tax and royalty costs are ignored. The flow rates are modelled as being pressure dependent. Iyer et al. [1998] present a MILP model where the decision variables are the choice of reservoirs to develop, selection from among candidate well sites, the well drilling 23

35 and platform installation schedule, capacities of well and production platforms and production rates for each time period. The flow rates are modelled as being pressure dependent, and the non-linearity reservoir performance is handled by piecewise linear interpolation. Drilling rig availability constraints are also taken into account. The model is solved by a sequential decomposition approach, although the suggested algorithm does not guarantee optimality. Jonsbråten [1998] develops a MIP model with the same decision variables as Haugland et al. However, unlike the previously mentioned articles, price uncertainty is taken into account, leading to a stochastic model. A two-dimensional reservoir model is used. A progressive hedging algorithm is used to find lower bounds for the problem, which converted to an implementable solution. van den Heever and Grossmann [2000] present a multiperiod MINLP model. Decision variables include investment in production platforms, well platforms and wells, well drilling schedule and well production profiles in each time period. The non-linear reservoir performance, including reservoir pressures, gas to oil ratio and cumulative gas produced, is expressed directly as non-linear functions of the cumulative oil produced. Aggregation/disaggregation as well as dynamic programming is used in the solution algorithm. van den Heever et al. [2000] develop a MINLP model. Their focus is on complex economic specifications such as taxes, tariffs and royalties. Although taking this into account leads to a better solution, the solution time increases by over an order of magnitude. This limits the size of the problems this model can handle. van den Heever et al. [2001] continue to investigate MINLP models with complex economic specifications. A Lagrangean decomposition approach with subgradient updating of multipliers leads to more than an order of magnitude decrease in solution time. The results from this model are still significantly better than from models not taking the complex objectives into consideration. Goel and Grossmann [2004] present a stochastic MILP model. The amount of petroleum reserves in the reservoir is considered to be uncertain. The stochastic solution approach leads to significant improvements in the NPV compared to the deterministic approach. Goel et al. [2006] present a stochastic MILP model. Investment decisions include selecting which well platforms, production platforms and pipeline connections should be installed in each time period, and the capacities of the well platforms and production platforms. Production rates in every time period is also to be decided. Both the size and deliverability of the field are considered to be uncertain. The model is solved by a Lagrangean duality based branch and bound 24

36 algorithm. The solutions are significantly better than those obtained in Goel and Grossmann [2004]. Barnes and Kokossis [2007] develop a deterministic MILP model that is quite different from those previously mentioned. Their objective is to minimize the CAPEX required to meet a target production profile. Earlier work has concentrated on maximizing profit, e.g NPV. The resulting model is decomposed into two parts: selecting an optimum drilling centre and optimizing the well selection. This reduces the complexity in solving the model. Erlingsen and Strat [2010] present a deterministic MILP model for planning of petroleum production from several fields in an area. The objective is to maximize total NPV before tax. Decision variables include number of wells and manifolds, installations at the production site, choice of transport alternatives from the production field to the customer and how to provide energy to the site installations. Production rates and capacities are also determined, as well as possible cooperation between different fields. Methods used to solve the model include Branch and Bound, Dantzig-Wolfe decomposition and Branch and Price. 25

37 2 Theory In this chapter the theoretical background for the mathematical model developed in this thesis will be presented. First, a short overview is given over linear programming (LP), integer programming (IP) and nonlinear programming (NLP). Then mixed integer nonlinear programming (MINLP) and mixed integer linear programming (MILP) are discussed. Finally, the branch and bound (B&B) algorithm is explained. 2.1 Mathematical programming models Linear programming Linear programming models are mathematical programming (MP) models where the objective function and constraints involve linear expressions. LP models are much easier to solve than NLP models. The reason for this is that the optimal solution of an LP model will always be at a vertex of the feasible region. If the form of the model is such that alternative solutions exist, an optimal solution will still be found at a vertex. The famous simplex algorithm only examines vertex solutions, rather than the generally infinite set of feasible solutions Integer programming If one or more of the variables in the model needs to take an integer value, the resulting model is an integer programming model. If all the variables are integer variables, the resulting model is a Pure Integer Programming (PIP) model. If both integer and continuous variables are used, the model will be a Mixed Integer Programming (MIP) model. IP can be used to model a wide range of practical problems. There are two main applications that involve the use of IP. The first application occurs in situations where it is only meaningful to make an integral amount of goods (cars, houses, ships etc.) or use an integral amount of resources(employees, machines etc.). IP can be used to model these decisions, although it can sometimes be good enough to use LP and round off the optimal solution values to the nearest integers. However, the main use of IP is to model decision variables which can only have the values 0 or 1. These variables model decisions where there is a clear yes or no answer. You either buy a car or you do not. This is a powerful way of modelling that is widely used. 26

38 IP models are in general much harder to solve than LP models. An IP model usually involves many times as much calculation as a similarly sized LP model due to the integer variables. Unlike even large LP models with thousands of variables and constraints which can almost certainly be solved within reasonable time limits, the same cannot be said for IP models. Several methods exist to solve IP models, of which cutting planes methods, enumerative methods, pseudo-boolean methods and Branch and Bound methods can be mentioned. The Branch and Bound method will be explained in Chapter Non-linear programming The difference between linear programming and non-linear programming is quite simple. If the objective function and/or at least one constraint contain a non-linear expression, the resulting model will be a NLP model. These are, as IP models, much harder to solve than LP models. NLP models often arise in both economical and engineering situations. For example, there is often increasing or decreasing returns to scale for the profit or costs of a company. Several factors give rise to the increased difficulty of solving an NLP model than an LP model. One of the problems is that the optimal solution is not necessarily found on a vertex of the feasible region, but may be found in the interior of the feasible region. When a solution has been found, it is also often difficult to determine whether it is a local or a global optimal solution Mixed integer non-linear programming If a mathematical model contains both integer variables and non-linear constraints or objective function, the resulting model is a mixed integer non-linear programming (MINLP) model. Such models share the characteristic features from IP and NLP models. This means that MINLP models are in general very hard to solve, and quickly become intractable. To reduce the amount of computer time that is necessary to solve a model, or to be able to solve it at all, MINLP models are often converted into mixed integer linear programming (MILP) models. This is done by linearising the non-linear functions by the use of some conversion method Mixed integer linear programming MIP models can be categorized as MILP or MINLP models. MILP models are those models which consist of linear expressions of integer and continuous variables 27

39 in all the constraints and the objective function. These are harder to solve than pure LP problems, but usually easier to solve than MINLP problems. Sufficiently small MILP problems can often be solved by a brute force approach, enumerating all possible solutions and choosing the best of these. As the problem size grows, this solution approach quickly becomes practically unsolvable, and special algorithms should be used to solve the problem. 2.2 Conversion of MINLP problems to MILP problems As MILP problems are usually easier to solve than MINLP problems it is often appealing to use a MILP formulation of the problem. Care should be taken to ensure that this will result in a sufficiently good representation of the system that is being modelled. LP formulations are usually much easier to solve, but the linearity assumption must be warranted. The easiest way of linearising a non-linear function is by fitting a linear function to the function values in the domain of the variable. A crude approach of doing this would be to linearise between the function values of the maximum and minimum values of the variable. A better linearisation can be found by linear regression of several function values. This will result in the single linear expression that diverts the least from the non-linear function. If the non-linear function doesnõt divert too much from this linearisation, this might be a successful approach that still gives a good representation of the reality whilst reducing the complexity of the model. If however the non-linear function is very non-linear for reasonable values of the variables, another approach must be used to ensure a sufficiently good representation of the reality. Piecewise linearisation of the non-linear function in question can often be a successful approach to achieve greater accuracy of the linearisation. Several breakpoints are defined on the non-linear function curve. The non-linear function is linearised between these breakpoints. Function values between the breakpoint values are found by linear interpolation between the breakpoint values. The advantage with this approach is increased accuracy in the modelling, compared to using a single linear function that should give a reasonably good representation for all possible values. The calculation time will however increase compared to the single linear function approach. Both the accuracy and calculation time depends on the number of breakpoints. More breakpoints will increase accuracy, but lead to longer calculation time. 28

40 2.3 Special ordered sets Special ordered sets is a concept that was introduced by Beale and Tomlin in 1979 [Beale and Tomlin, 1979]. A special ordered set of type 1 (SOS1) is a set of variables (continuous or integer) within which exactly one variable must be nonzero. A special ordered set of type 2 (SOS2) is a set of variables within which at most two can be non-zero. The two variables must be adjacent in the ordering given to the set [Williams, 2008]. Although it is possible to model the restrictions that a set of variables belongs to an SOS1 set or an SOS2 set using integer variables and constraints, it is however great computational advantage to be gained from treating these restrictions algorithmically [Williams, 2008]. The most common application of SOS1 sets is to modelling what would otherwise be 0-1 integer variables. The most common application of SOS1 is to modelling non-linear functions. A chain of linked SOS sets can be used to model non-separable functions without converting the model into one where the non-linearities are all functions of a single variable. Doing so may reduce the computational difficulty. 2.4 Branch and bound Branch and bound search is a popular algorithm used to efficiently find optimal solutions for IP problems, including MILPs. The idea is to determine for large classes of solutions whether they are likely to contain optimal solutions or not, and doing so without explicit enumerations of all its members. Only the most promising classes have to be searched in detail. This section is largely based on Ronald L. Rardin s textbook on optimization [Rardin, 2000]. Branch and bound algorithms form classes of solutions and analyze whether these classes can contain optimal solutions by analyzing associated relaxations. More detailed enumeration ensues only if the relaxations fail to be definitive. Branch and bound searches through partial solutions, which are solutions that have some decision variables fixed, while other are free or undetermined. A branch and bound search starts at an initial or root partial solution where all the variables are free. A partial solution is terminated in a branch and bound search if it either identifies a best completion or prove that none can produce an optimal solution in the overall model. If a partial solution cannot be terminated in a branch and bound search it is branched by creating two subsidiary solutions by fixing a previously free binary variable. One of the subsidiary partial solutions 29

41 is the same as the current solution except that a binary variable is fixed to be 0, and the other partial solution fixes the same variable to be 1. It is important to know how a search stops. A branch and bound search stops when every partial solution in the tree has been either branched or terminated. As long as some partials solutions remain, the branch and bound search must select an active partial solution to explore next. Several schemes to decide which partial solution to explore exist. The simplest scheme is called depth first. A depth first search selects at each iteration an active partial solution with the most components fixed. This means the "deepest" partial solution in the search tree, giving the name to this search scheme. Branch and bound searches often try to take advantage of various relaxation methods. First the incumbent solution at any stage in a search is defined to be the best feasible solution known so far. The candidate problem to a partial solution is the restricted version of the model obtained when variables are fixed as in the partial solution. With these two properties defined, several results can be obtained. First, it s known that the feasible completions of any partial solution are the feasible solutions to the corresponding candidate problem, and thus the objective value of the best feasible completion is the optimal objective value of the candidate problem. Also, if any relaxation of the candidate problem is infeasible, then the related partial solution can be terminated because it has no feasible completions. Then, if any relaxation of a candidate problem has optimal objective value no better than the current incumbent solution, the associated partial solution can be terminated because no feasible completion can improve on the incumbent solution. Thirdly, if an optimal solution to any constraint relaxation of a candidate problem is feasible in the full candidate, it is a best feasible completion of the associated partial solution. After checking whether a new incumbent has been discovered, the partial solution can be terminated. The three properties described in the former paragraph provide powerful concepts that can be used to efficiently discard large amount of possible combinations of the variables at once. An example on this is the LP-based branch and bound search. This class of branch and bound search algorithms branch by fixing an integer restricted decision variable that had fractional value in the associated candidate problem relaxation. If several integer variables have fractional value the algorithm will usually branch by fixing the variable that s closest to an integer value. In addition to the depth first scheme several other schemes to choose the next partial solution to explore. An example of this is the best first search, that at each iteration selects an active partial solution with the best parent bound. The parent node is the branched node. Parent bounds can also be used to terminate 30

42 solutions. Whenever a new incumbent solution is discovered, any active partial solution with parent bound no better than the new incumbent solution value can immediately be terminated. Other schemes include the depth forward best back scheme, and the nearest child rule. Parent bounds can also be used to estimate the error by accepting the incumbent solution. At any stage of a branch and bound search, the difference between the incumbent solution value and the best parent bound of any active partial solution shows the maximum error in accepting the incumbent as an approximate optimum. Thus you know how far from optimum you can maximum be by accepting the incumbent solution, and stopping the search there. 31

43 3 Mathematical model In this chapter a new mathematical model for optimization of offshore natural gas field development will be presented. First the main assumptions and simplifications are given, before the model is presented stepwise through the natural gas value chain. The model follows a naming convention. Small letters are used for variables, and capital letters are used for constants. Indexes are small letters in subscript. Capital letters in subscript are part of variable or constant names. 3.1 Assumptions and simplifications The model is developed for petroleum deposits that consist mainly of natural gas. Most natural gas fields will produce some liquids i.e. natural gas liquids (NGL), condensate or crude oil together with the natural gas. It is assumed that these liquids are produced as a constant fraction of the natural gas production. This is a simplification of the realities, as the liquids production will usually vary to some degree as the pressure in the reservoir declines. The need to develop a transport alternative and capacity to be able to sell these products is not explicitly formulated in the model. It can be taken into consideration to some degree by increasing the cost of the production facilities and transport alternatives for natural gas. It is assumed that enough exploration activities have been done to reveal the main properties of the natural gas reservoir that is to be developed. Especially this includes the initial gas resource in place, pressure response to production and how productive the reservoir is probable to be. The model is deterministic, not stochastic. This means that all values are assumed to be known with certainty. Indeed, this is a major assumption as most natural gas fields are affected to various degree by uncertainty. Most parts of the model will in reality have some degree of uncertainty related to them. The reservoir can be smaller or larger than what it is initially believed to be. The investment cost of the different parts of infrastructure is also assumed to be known. It is however not uncommon that field developments become more expensive than originally planned. An overview by the Norwegian Petroleum Directorate (NPD) in 2009 showed that field developments then were on average 13.8 per cent more expensive than when the plan for development was delivered. Furthermore, oil and gas prices that are used in the model are projected for the whole field life time which can be several decades. With oil prices rising above 140 dollar, falling below 40 dollar and rising towards 130 dollar in the course of three years, it is evident that projecting 32

44 the prices 50 years ahead will be close to impossible. It is assumed that there will always be additional demand for the natural gas in the market, and that the amount of natural gas supplied from the field will not have an influence on the market price. The model allows for a long term contract to be agreed with a customer, resulting in a higher price in return of meeting a given demand for several years. The model that will be presented is a MILP model. It is assumed that costs that are given as constants are in fact constant, that linear relations used are in fact linear, and that linearisation of the non-linear functions gives a sufficiently good representation of the realities. Furthermore, the model is developed with strategic decisions in mind, as opposed to the operational and tactical decisions. It is intended to be used in the early phase of offshore natural gas field developments to give an indication of what might be an optimal approach to the particular natural gas field in question. The user is assumed to be in position that has the power to make sure that the suggested decision from the model might be realised. The time frame of the field development is divided into a given number of periods. All investment and production decisions are made at the start of a given period. This means that the reservoir pressure and flow rates are assumed to be constant within a time period. This is a simplification of the real conditions, as the reservoir pressure and thus the production potential of a natural gas reservoir will decline during the production in a time period. Thus the possible production in a time period will usually be slightly overestimated in the model. It will however be slightly counteracted by the fact that the reservoir pressure in following periods will be underestimated, as the additional production in previous periods will lead to additional decline in reservoir pressure. The model consists of seven main parts, each taking into account a particular aspect of the natural gas value chain. These main parts are the reservoir model, wells, flowline infrastructure, production infrastructure, energy infrastructure, transport infrastructure and customers. The relationship between these can be illustrated with Figure 3.1. The objective of the model is to find the maximum net present value (NPV) of an offshore natural gas field development project, pre-tax. Although taxes and royalty schemes will in reality affect the optimal solution, it is chosen to disregard this in the model. The reason for this is that the tax regime of a petroleum producing region will often change during a field development s life time. As recently as the spring of 2011, the oil tax rate was raised in the United Kingdom, apparently 33

45 Energy Reservoir Wells Flowlines Production Transport Customer Figure 3.1: Mathematical model structure surprising oil companies and putting field developments into jeopardy. 3.2 Reservoir model The first part of the model is the reservoir model. This part of the model is very important, as the revenue of a field development project is dependent on how much natural gas that may be produced at a given time. The reservoir is modelled as a pressurised tank, with a set of characteristic properties. G 0 - initial gas resource in place P R0 - initial reservoir pressure P Z0 - initial p/z factor (pressure divided by compressibility) P IND - productivity index P ZDROP - reduction in p/z factor by producing one unit of gas L V - liquids produced with one unit of gas P DMAX - maximum pressure differential between reservoir and wellbore flowing pressure The field development will happen over some time frame. Instead of having a continuous time variable, the time frame will be divided into several discrete time periods. This introduces the need to define a time index, as well as a set of time periods. The number of time periods will vary from one field development to another. t - index for time period t T - set of all time periods The gas flow from the reservoir in each time period is one of the main outputs from the model. q t - natural gas flow from the reservoir in time period t Having defined these parameters, the reservoir model can be developed. First, an obvious constraint is that the total gas production from the natural gas field 34